Parallel coordinates

The technique is well-suited to predicting the behaviour of a multivariable controller, even before steptesting has been started. Plant history databases comprise a number of instrument tag names with measurements collected at regular intervals. If we imagine the data arranged in a matrix so that each column corresponds to either a MV or a C V in the proposed controller and each row is a time stamped snapshot of the value of each parameter. To this we add a column in which we place the value of the proposed MVC objective function (C) derived from the values in the same row (where P are the objective coefficients for the m CVs, and Q the objective coefficients for the n MVs), i.e. 

Each row in the database is then plotted as a line on the parallel coordinates chart. The result will initially appear very confused with a large number of lines superimposed. The next step is to add the HI/LO constraints on each vertical axis. If a line violates any constraint on any axis then the whole line is deleted. The lines remaining will each represent an occasion in the past when all the process conditions satisfied the constraints. The final step is to choose the line for which the value on the cost axis is the lowest. Since this axis is the MVC cost function, the line with the lowest value will represent the operation that the MVC would select. 

Provided that the process has at some stage operated close to the optimum (as defined by the MVC) then the chosen data set will give some idea of the operating strategy that the MVC will implement. If different firom the established operating strategy, this approach 

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As has been stated above the lines for equal values of the ultrasonic sound velocity of mineral oil fractions in the log v -(n — d) graph are straight. Therefore it is possible to construct a nomogram with the parallel coordinates log v% and (n — 0.181 d), the value of 0.181 in the function (n — 0.181 d) being somewhat more accurate than in the function (n — d). 

Figure 14. Parallel coordinate plots of the dihedral angles versus the dihedral sequence that display the extent of the sampling of the conformational phase space of Ala-ciiPro-Tyr, in 2 ns free MD simulations (left hand plots) and in 2 ns biased MD simulations (right hand plots). The top plots display all the conformations sampled during a MD trajectory starting from the bl conformation, and the bottom plots, the ones sampled during a simulation starting from the al conformation (see Figure 3).

Inselberg, A., and Dimsdale, B. (1990). Parallel coordinates A tool for visualizing multi-dimensional geometry. In Proceedings of IEEE Conference on Visualization, A. Kaufman (Ed.). San Francisco, CA IEEE Computer Society Press, pp. 361-378. 

In some cases, it was deemed impossible to optimize the molecule to reach all desired properties. In particular, the concept of competing objectives is well established in multicriteria optimization case studies. Competing objectives are those where an improvement in one objective results in the deterioration in the other. Parallel coordinate plots, a simple tool that can be used to identify competing objectives, were found to be a quick and useful aid. Figure 8.17 illustrates the case of two competing objectives, solubility and in vitro enzyme potency the crossed lines clearly identify the objectives are in competition. 

FIGURE 8.17 Parallel coordinate plots allow detection of competing objectives in optimization processes. This is an extreme example, where solnhdity and in vitro potency ( Primary XC50 ) are competing. When solnhdity increases, potency decreases and vice versa. This results in the characteristic erossed-hne plot shown here. 

FIGURE 9A (a) Parallel coordinates plot of Project A compounds six most important experimental properties (normalized fiom 0 to 1) and weighted desirability score. Higher scores are depicted in green, (b) Parallel coordinates plot of the six compounds having a weighted desirability score above 0.8. The actual candidate of Project A is highlighted in red and has a score of 0.83. For color details, please see color plate section. 

FIGURE 9.4 (a) Parallel coordinates plot of Project A compounds six most important... 

Couette, and rotational parallel coordinates coordinates measuring thickness is... 

Figure 26 Views of the crystal structures of the dual host crown ether complexes of 50 with (a) NaNOs and (b) NaCl, iUnstrating the antiparallel and parallel coordination modes observed for the two guests, respectively.

Parallel coordinates is two-dimensional graphical method for representing multiple dimensional space. In the example shown in Figure 8.13, a point in seven-dimensional space is represented by the coordinates (xi, X2, X3, X4, X5, Xg, X7). Since we cannot visualise space of more than three dimensions, the value of each coordinate is plotted on vertical parallel axes. The points are then joined by straight lines. 

While there is yet to be developed an entirely satisfactory solution to this problem, some ideas have been applied successfully. One is the use of a radar plot. This is similar to parallel coordinates except that the axes are arranged radially. Only a limited number of CVs and MVs are practicable - perhaps a maximum of around 12, so only the more important variables are included. 

Chernoff Faces, Andrews Curves, Star Diagrams, and Parallel Coordinate Plots... 

Wegman, E., Hyperdimensional data analysis using parallel coordinates, J. Ann. Statist., 41 (1970) 457-471. 

Inselberg, A., The plane with parallel coordinates. The Visual Computer, 1 (1985) 69-91. 

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